Real numbers are numbers that can be represented by some infinite or finite decimal representation, such as 0.707106781187...
The term "real number" exists for historical reasons; it was once thought that such numbers "actually" existed (hence the word "real"), in contrast to the supposed non-existence of imaginary numbers. However, advances in complex analysis in the 19th century revealed that imaginary numbers do, in fact, exist, and the terminology of "real" and "imaginary" is now understood to be just that--terminology.
In a number line representation, the real numbers correspond to all the points on a geometric line. The distance between any two points on a line is a real number.
The real numbers include within them all of these other kinds of numbers:
- The "natural numbers" or positive integers, 1, 2, 3, ...
- Zero and the negative integers
- Fractions, like 355/113
- Any decimal representation which terminates (comes to an end), like 6.023, because this is just a way of writing a fraction (in this case, 6023/1000)
- Any decimal representation which repeats or recurs, like 1.86292929292929..., because these can be shown to be fractionsTemplate:Prove
- Irrational numbers, like π = 3.1415926525..., whose decimal representations never repeat or terminate.
Formally, real numbers are defined as the unique field which is ordered, complete, and Archimedean. The reals can be constructed from the rationals by means of Dedekind cuts or Cauchy sequences, i.e. it is the completion of the metric space of rational numbers.
The real numbers do not include or (infinity and minus infinity). However, there are non-standard models of real numbers which include or include both and .
There is no largest real number, because you can always make a real number larger by adding 1 (or 137.035 or 6.023·1023) to it, and no smallest real number, because you can always make a real number smaller by subtracting from it.
Every real number is finite. One way to see this is to observe that you cannot subtract infinity from itself—the result is indeterminate—but, for any real number x, then x - x = 0, exactly.
It is sometimes convenient to have a set of numbers that does include infinity. For example, in computer programming, "real arithmetic" is often done by a specific system defined by standard IEEE 754-1985; this system is built in to modern processor chips. It provides for values which print out as INF and -INF and which participate in arithmetic as if they were numbers. Thus, division by zero, which was often an error that stopped calculation on older machines, can be a legal operation which simply produces a +INF or -INF result. The system of numbers implemented in IEEE 754 is known in mathematics as the "affinely extended real numbers."
Notes and references
- In reality, the actual values the computer uses are very-high-precision fractions which can equal or approximate real numbers